
Furthermore, if \(u\)  and \(v\)  are
sufficiently smooth in \(\xi \)  (e.g. they are Hölder continuous [1]})
then we may write
\(u(\xi , t)= \sum _{k=-\infty }^\infty \hat{u}(k, t) \textrm {e}^{\textrm {i}\pi k \xi },\qquad v(\xi , t)= \sum _{k=-\infty }^\infty \hat{v}(k, t) \textrm {e}^{\textrm {i}\pi k \xi }. \) 
